Topological group criterion for in compact-open-like topologies, II

We continue from “part I” our address of the following situation. For a Tychonoff space Y, the “second epi-topology” σ is a certain topology on C(Y), which has arisen from the theory of categorical epimorphisms in a category of lattice-ordered groups. The topology σ is always Hausdorff, and σ interacts with the point-wise addition + on C(Y) as: inversion is a homeomorphism and + is separately continuous. When is + jointly continuous, i.e. σ is a group topology? This is so if Y is Lindelöf and Čech-complete, and the converse generally fails. We show in the present paper: under the Continuum Hypothesis, for Y separable metrizable, if σ is a group topology, then Y is (Lindelöf and) Čech-complete, i.e. Polish. The proof consists in showing that if Y is not Čech-complete, then there is a family of compact sets in βY which is maximal in a certain sense.